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mick25

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This isn't homework but I'm having trouble understanding the concept of non-homotopic and homotopic Jordan curves.

My understanding of Jordan curves and homotopy:

This is an example from my lecture notes:

Let A = C\<0>; ie. A is the complex plane with a "hole" at the origin

There are precisely 3 non-homotopic Jordan curves in A.

thanks Office Shredder

My understanding of Jordan curves and homotopy:

A Jordan curve is a simple closed curve (ie a closed curve that only intersects at the endpoints; f(z1)=f(z2) => z1=z2) such that its exterior is a disjoint union of two open sets:

i) the "inside" (of the J. curve) is a bounded connected open set

ii) the "outside" (of the J. curve) is an unbounded connected open set

iii) the said Jordan curve is the boundary of BOTH "inside" and "outside" (not true for general curves)

Two closed curves, f1, f2 are homotopic in a connected open set A in C when we can continuously deform with closed curves fk; k in [0,1].

i) the "inside" (of the J. curve) is a bounded connected open set

ii) the "outside" (of the J. curve) is an unbounded connected open set

iii) the said Jordan curve is the boundary of BOTH "inside" and "outside" (not true for general curves)

Two closed curves, f1, f2 are homotopic in a connected open set A in C when we can continuously deform with closed curves fk; k in [0,1].

This is an example from my lecture notes:

Let A = C\<0>; ie. A is the complex plane with a "hole" at the origin

There are precisely 3 non-homotopic Jordan curves in A.

**Here's my confusion:**thanks Office Shredder

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